Journal of Physics A: Mathematical and Theoretical

J. Phys. A: Math. Theor. 53 (2020) 415201 (21pp) https://doi.org/10.1088/1751-8121/abad77

Classical and quantum complexHamiltonian curl forces

M V Berry

H H Wills Physics Laboratory, Tyndall Avenue, Bristol BS8 1TL, United Kingdom

Received 23 June 2020, revised 29 July 2020Accepted for publication 7 August 2020Published 10 September 2020

AbstractA class of Newtonian forces, determining the acceleration F(x, y) of particlesin the plane, is F=(Re F(z), Im F(z)), where z is the complex variable x + iy.Curl F is non-zero, so these forces are nonconservative. These complex curlforces correspond to completely integrable Hamiltonians that are anisotropicin the momenta, separable in z and z∗ but not in x and y if the curl is nonzero.The Hamiltonians can be quantised, leading to unfamiliar wavefunctions, evenfor the (non-curl) isotropic harmonic oscillator. The formalism provides analternative interpretation of the analytic continuation of one-dimensional realHamiltonian particle dynamics, where trajectories are known to exhibit intricatestructure (though not chaos), and is a Hermitian alternative to non-Hermitianquantisation.

Keywords: non-Hermitian, PT symmetry, integrable

(Some figures may appear in colour only in the online journal)

1. Introduction

Curl forces [1] are Newtonian forces describing accelerations of particles. They depend onposition (but not velocity), and are not the gradient of a potential, so their curl is not zero:

r = F (r) , ∇ × F = 0. (1.1)

These forces are non-conservative, because the work∮

dr · F (r) done while transporting theparticle around a circuit is non-zero, and equal to the flux of ∇ × F through the circuit; never-theless, they are non-dissipative, because volume in the state space of position and velocity isconserved. For most curl forces, there is no associated Hamiltonian or Lagrangian, and there-fore no straightforward quantisation. But for a zero-measure subset of curl forces an underlying

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1751-8121/20/415201+21$33.00 © 2020 The Author(s). Published by IOP Publishing Ltd Printed in the UK 1

J. Phys. A: Math. Theor. 53 (2020) 415201 M V Berry

Hamiltonian does exist [2, 3] and can be quantised. In such Hamiltonians, which may or maynot be integrable [4], the kinetic energy is anisotropic in the canonical momenta.

My purpose here is to explore a subset of this Hamiltonian subset of curl forces, and theirquantisation. These forces are characterised by an underlying complex structure. They describea particle moving in the plane r = (x, y), and F(r) depends only on the complex variable z = x+ iy through a scalar function F(z). For such ‘complex curl forces’, (1.1) reduces to dynamicsin the complex plane:

z = F (z) , (z = x + iy) . (1.2)

The function F(z) is assumed to be analytic in the z plane, possibly with isolated pole or branch-point singularities.

Complex classical dynamics of this type has been extensively studied from a differentviewpoint [5]. Starting from the one-dimensional Hamiltonian

H (x, p) =12

p2 + U (x) , (1.3)

with real x, p, in which U(x, y) need not be real, the variable x has been extended to the complexplane. A series of remarkable researches [6–11] has revealed that even for simple potentialsU(z) the solutions of (1.2) represent exquisitely intricate trajectories.

The additional perspectives provided here are that the complex dynamics (1.2) has the fol-lowing features: it represents a special case of curl force dynamics; it is generated in the realplane x, y by a real and completely integrable two-freedom real Hamiltonian; and it can bequantised in the real plane. Section 2 describes the classical Hamiltonian, using two differentrepresentations, one of which is separable for any curl force F(z). Section 3 describes the cor-responding Hermitian quantisation. Section 4 explores some examples; although simple, thesepossess unanticipated subtle features. This study raises questions that point to directions forfurther study; some are listed in the concluding section 5.

We are here considering real x and y and also real t, but note that when t is complexified thedynamics has additional richness. One motivation for this is the motion of electrons in crystalswhen a magnetic field is applied [12]. Then the force depends on velocity as well as position,but the dynamics (1.2) is still relevant because the velocity dependence can sometimes beeliminated by transforming to a complex time variable that may be periodic [13] as a functionof physical (real) time. The associated return maps display intricate fractal structure [14].

To avoid possible confusions, I note that the class of curl forces (1.2) is different fromthe curl force (a.k.a. ‘scattering force’) exerted on a small polarisable absorbing particle byan optical field ψ(r) [3, 15], even when this is has the form ψ(x + iy). This optical force isF (r) = Im [ψ∗ (r)∇ψ (r)], which is different from (1.2) except for the case of a single opticalvortex, for which ψ = x + iy and F(z) = iz in (1.2). Second, I will not discuss non-Hermitianquantisation, with its focus on PT symmetry, about which great deal is already known [16].

2. Classical Hamiltonian curl force formalism

It is easy to check that for the complex dynamics (1.2) the curl vector in (1.1) is

(∇ × F)⊥ ≡ C (r) = 2 Im F′ (z) , (2.1)

so all these complex forces have non-zero curl, except for the trivial case F(z) = ±z + constant.

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J. Phys. A: Math. Theor. 53 (2020) 415201 M V Berry

To display the Hamiltonian structure underlying (1.2), the first step is to define the natural‘potential’, satisfying Laplace’s equation,

F (z) = −U′ (z) , i.e.U (z) = −∫ z

0dζF (ζ) . (2.2)

Then (1.2) implies the complex conserved ‘energy’

12

z2 + U (z) = Ec, (2.3)

which determines the velocity (up to a sign representing time-reversal); it follows thattrajectories with the same Ec cannot cross [6].The trajectories z(t) for any U(z) and specified Ec are the integral curves of the velocity vectorv = {Re z, Im z}, and can be obtained formally by quadrature:

t − t0 =

∫ z(t)

z0

dz√2 (Ec − U (z))

. (2.4)

This conceals more than it reveals, because of the complications introduced by the Riemannsurface structure [14] associated with the branch cuts and inverse functions required to deter-mine the trajectory. An important role is played by the stagnation points [6] defined by U(z) =Ec, where the velocity vanishes.

The real and imaginary parts of Ec correspond to two real conserved quantities:

12

(x2 − y2) + Re U (x + iy) = E, xy + Im U (x + iy) = K (2.5)

either can be defined as a Hamiltonian generating (1.2). We choose the Hamiltonian corre-sponding to the first conservation law:

H = E =12

(p2

x − p2y

)+ Re U (x + iy) , i.e. px = x, py = −y. (2.6)

The second conservation law then corresponds to the additional conserved phase-spacefunction

−px py + Im U (x + iy) = K. (2.7)

(The alternative Hamiltonian defined by the second conserved quantity in (2.5) is + px py +Im U (x + iy), and the association between velocities and momenta is different: x = py, y =px .) The existence of a second integral of motion implies that the Hamiltonian (2.6) (or (2.7))is completely integrable (though usually nonseparable) [14]: therefore in the dynamics (1.2),there is no chaos. (The anisotropic Kepler problem [17, 18], describing electrons in semicon-ductors and familiar in quantum chaology, is an example of a Hamiltonian curl force [2]; butit does not belong to the class (2.6) because its potential U(x, y) = −1/√(x2 + y2) is not thereal part of a function of x + iy; moreover it is nonintegrable.)

To prepare for quantisation, it is convenient to express the Hamiltonian (2.6) in terms of newphase space variables, with coordinates z and z∗, regarded as independent coordinates, and theconvenient notation

X = z = (x + iy) , Y = z∗ = (x − iy) ,

PX =12

(px − ipy

), PY =

12

(px + ipy

).

(2.8)

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J. Phys. A: Math. Theor. 53 (2020) 415201 M V Berry

Then (2.6) becomes the alternative Hamiltonian

H =

(P2

X +12

U (X))

+

(P2

Y +12

U∗ (Y))

= E = EX + EY , (2.9)

which is obviously separable, with energies EX, EY corresponding to the two freedoms in thisrepresentation. The second conserved quantity (2.7) transforms to

iK =

(P2

X +12

U (X))−(

P2Y +

12

U∗ (Y))

= + (EX − EY) , (2.10)

which is simply an awkward way of stating that the X and Y Hamiltonians are conserved sep-arately. It is worth emphasising that this complexification is different from the more familiarprocedure in which coordinates and momenta are mixed [19].

3. Quantum complex curl force formalism

The Hamiltonian (2.6) can be quantised in the familiar way, using px = −i∂x, py = −i∂y,leading to a Schrödinger equation involving a Hermitian operator:

((−∂2

x + ∂2y

)+

12

Re U (x + iy))ψ (r) = Eψ (r) . (3.1)

The different signs in the x and y derivatives are a characteristic feature of complex curl forcequantisation, and will have unexpected consequences. With the exception of the free particleand linear potential, to be considered in sections 4.1 and 4.2, the potential Re U (x + iy) is notseparable in x and y.

However, the form (2.9) is separable, and the substitution PX = −i∂X , PY = −i∂Y leads tothe Schrödinger equation

(−∂2

X − ∂2Y +

12

U (X) +12

U∗ (Y))Ψ (X, Y) = (EX + EY)Ψ (X, Y) = EΨ (X, Y) , (3.2)

whose solution is any superposition of the product wavefunctions

Ψ (X, Y) = Ψ ((x + iy) , (x − iy)) = ψX (X)ψY (Y) , (3.3)

where ψX and ψY satisfy(−∂2

X +12

U (X))ψX (X) = EXψX (X) ,

(−∂2

Y +12

U∗ (Y))ψY (Y) = EYψY (Y) .

(3.4)

It is easy to check directly that Ψ(r) = Ψ(X(x, y), Y(x, y)) is a solution of (3.1). Thus is estab-lished the quantisation of the classical complex curl forces defined by (1.2). The energy is E= EX + EY . We will mostly consider E real; EX and EY need not be real, and then Im EX =−Im EY can be interpreted as ψX describing gain or loss, with ψY describing a compensatingloss or gain.

The complex-separated Hamiltonian in (3.2) is Hermitian, but the separate X and Y contri-butions in (3.4) are non-Hermitian Hamiltonians. The separate Schrödinger equations in (3.4),are the same as the analytically continued one-dimensional operators studied in non-Hermitian

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J. Phys. A: Math. Theor. 53 (2020) 415201 M V Berry

quantum mechanics [16]. These studies, particularly where U (X) is non-Hermitian and PTsymmetric, have led to ingenious interpretations, involving waves decaying in particular sec-tors of the x, y plane, leading to a discrete spectrum of real energies. Here the different focusis on dynamics and quantisation of curl forces in the full real x, y plane.

4. Simple examples

4.1. Free particle

When U(z) = 0, the solutions of (3.1) are

ψ (x, y) = exp(i(kxx + kyy

)), E =

12

(k2

x − k2y

), (4.1)

which are propagating plane waves for kx and ky real. The solutions of (3.2) in the separatedform (3.3) are

Ψ (X, Y) = exp (i (KXX + KY Y)) = exp (i (KX + KY ) x) exp ((KY − KX) y) ,

E = K2X + K2

Y ·(4.2)

which are evanescent or growing waves for KX and KY real. To recapture the propagating planewaves (4.1), it is necessary to choose separated momenta that are complex (cf (2.8)):

KX =12

(kx − iky

), KY =

12

(kx + iky

). (4.3)

Perhaps surprisingly, evanescent plane waves (KX, KX real), can be expressed as superpositionsof propagating waves (kX, kX real) [20]; extensions of this idea will be used in the next twosections.

4.2. Linear potential

This is

U (z) = az, (a = a1 + ia2) , i.e. Re U (x + iy) = a1x − a2y, (4.4)

which according to (1.2) and (2.2), corresponds to the constant force F = −a, i.e. force vectorF = (−a1, −a2). For positive a1, a2, particles in the positive quadrant x > 0, y > 0 are attractedto the x and y axes. This simple example is not a curl force (cf (2.1)), because the force isconstant; nevertheless it is instructive to examine it in some detail.

The Schrödinger equation (3.1) is(

12

(−∂2

x + ∂2y

)+ a1x − a2y

)ψ (r) = Eψ (r) . (4.5)

This special case is separable in x, y for all a, and general solutions with energy E involvesuperpositions of linear combinations Ci1, Ci2 of the Airy functions Ai and Bi [21]:

ψ (x, y) = Ci1

((2a1)1/3

(x − Ex

a1

))Ci2

((2a2)1/3

(y − Ey

a2

)),

E = Ex − Ey·(4.6)

For fixed E, this is a one-parameter family of degenerate states, labelled by Ex or Ey.

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J. Phys. A: Math. Theor. 53 (2020) 415201 M V Berry

An unsurprising special case is the orthonormal eigenfunctions in the positive quadrant,decaying at infinity and with Dirichlet boundary conditions on the axes, namely

ψm,n (r) =

(4

a1a2

)1/6 Ai(

(2a1)1/3x − αm

)Ai

((2a2)1/3y − αn

)

Ai′ (−αm) Ai′ (−αn),

{m, n} = 1, 2, 3 . . . , Em,n =1

21/3

(αna2/3

2 − αma2/31

),

(4.7)

in which the positive numbers αn are the Airy zeros: Ai (−αn) = 0. The mode energies Em,n

are negative as well as positive (because of the signs in the derivatives in (3.1)), and real andnondegenerate for almost all a1, a2 (if a1 = a2, E = 0 is infinitely degenerate, and if a1 anda2 are proportional to Airy zeros some levels are pairwise degenerate). Figure 1(a) illustratesone of these modes. The corresponding classical trajectory (figure 1(b)) consists of segmentsof parabolas with accelerations −a1, −a2, specularly reflected from the axes, and bounded bya rectangular caustic. This is standard torus quantisation of an integrable quantum system.

For the general complex-separated form (3.3), the functionsψX andψY in the product wave-function Ψ(X, Y) can alternatively be expressed in terms of any linear combinations Ci of Aiand Bi:

ψX (X) = Ci1

((X − 2EX

a

)(12

a)1/3

)+ Ci2

((X − 2EX

a

)(12

a)1/3

),

ψY (Y) = Ci3

((Y − 2EY

a∗

)(12

a∗)1/3

)+ Ci3

((Y − 2EY

a∗

)(12

a∗)1/3

),

E = EX + EY ·

(4.8)

Here we encounter the first unfamiliar feature: these X, Y complex-separated states are verydifferent from those of the x, y real separation (4.6). This is illustrated in figure 1(c) for thecase where the Ci1 and Ci3 are Ai functions and Ci2 = Ci4 = 0 (i.e. Ψ is the product of twoAiry functions), for the same a and E and the same quadrant as ψm,n in figure 1(a). The twolines of isolated zeros correspond to points where the argument of each of the two Ai functionsis real and equal to any of the Airy zeros −αn, contrasting with the intersecting line zeros ofthe real modes ψm,n. The asymptotics of Ai and Bi [21], in particular the Stokes phenomenon[22], imply, in contrast to the exponential decay of the modes ψm,n in the positive quadrant,that for all choices of the Ci1. . .Ci4 there are sectors in each quadrant of the x, y plane whereΨ diverges exponentially [e.g. as O(exp(|r|3/2))] (see figure 1(c) where the wave rises to ∼e40

on the edge of the region shown).The two separations must be related, because both families of states are solutions of (4.5).

To illustrate the relation, we choose superpositions of the standard Ai functions, and forconvenience write the wavefunctions using Dirac state-vector notation. For (4.6), this is

⟨r∣∣ψ

(Ex , Ey

)⟩=

22/3

(a1a2)1/6 Ai(

(2a1)1/3(

x − Ex

a1

))Ai2

((2a2)1/3

(y − Ey

a2

)),

E = Ex − Ey·(4.9)

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J. Phys. A: Math. Theor. 53 (2020) 415201 M V Berry

Figure 1. (a) Mode ψ5,10(r) in (4.7), for a = 1 + 2i. (b) The corresponding classicaltrajectory, (c) complex-separated mode log |Ψ| in (4.8) with Ci = Ai, also with a = 1 +2i, and EX = 15i, EY = 5–15i.

States with different Ex, Ey are orthogonal, and the prefactor ensures the normalisation

⟨ψ(E1x, E1y

) ∣∣ψ(E2x, E2y

)⟩= δ (E1x − E2x) δ

(E1y − E2y

). (4.10)

This follows from [23]

∫ ∞

−∞dξ Ai (ξ − c) Ai (ξ − d) = δ (c − d) . (4.11)

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J. Phys. A: Math. Theor. 53 (2020) 415201 M V Berry

The states (4.9) are not square-integrable in the full x, y plane, but they are bounded everywhere.The unbounded corresponding complex-separated states (cf (4.8) are

⟨r∣∣Ψ (EX , EY)

⟩

= Ai

((12

a)1/3(

(x + iy) − 2EX

a

))Ai

((12

a∗)1/3(

x − iy − 2EY

a∗

)),

E = EX + EY ·

(4.12)

Note the different signs in the energies in (4.9) and (4.12).The relation between the two separations is that each of the states |Ψ⟩ in (4.12) can be

written as a superposition of the states |ψ⟩ in (4.9):

|Ψ (EX , EY)⟩ =

∫∫dEx dEy

⟨ψ(Ex, Ey

) ∣∣Ψ (EX, EY)⟩ ∣∣ψ

(Ex , Ey

)⟩. (4.13)

The overlap, derived in appendix A, is

⟨ψ(Ex, Ey

) ∣∣Ψ (EX, EY)⟩

=1

(a1a2)1/6 Ai

⎛

⎜⎝ − 21/6

(|a|2a1a2

)2/3

(a2

2Ex + a21Ey + ia1a2 (EX − EY)

)⎞

⎟⎠

× δ(EX + EY − Ex + Ey

).

(4.14)

The δ function confirms that any of the degenerate states |Ψ⟩ with energy E can be expressedas a superposition of the degenerate states |ψ⟩ with the same energy E.

The superposition relating the separations could equally be carried out in the positive quad-rant rather than the full x, y plane, with Ψ expressed as a sum over the discrete states ψm,n in(4.7). Althoughψm,n satisfies Dirichlet boundary conditions, Ψ does not. At the classical level,this corresponds to Ψ being the quantisation of a collection of trajectories of the torus type infigure 1(b), exploring rectangles of different side ratios.

It is worth remarking that for linear potentials the separations (4.9) and (4.12) are twomembers of a more general family, all satisfying (4.5):

Ψ (x, y; A, B) = Ai

⎛

⎝ (Ax + By)

(2 (Aa1 + Ba2)(B2 − A2

)2

)1/3

− E1

(2(A2 − B2

)

(Aa1 + Ba2)2

)1/3⎞

⎠

× Ai

⎛

⎝ (Bx + Ay)

(2 (Aa2 + Ba1)(B2 − A2

)2

)1/3

− E2

(2(A2 − B2

)

(Aa2 + Ba1)2

)1/3⎞

⎠ ,

(4.15)

with energy E = E1–E2. The separation (4.9) corresponds to A = 1, B = 0, and (4.12)corresponds to A = 1, B = i.

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J. Phys. A: Math. Theor. 53 (2020) 415201 M V Berry

Figure 2. Force vectors and their integral curves for the quadratic potential (linear force)(4.16), for (a) a = 1 (not a curl force), (b) a =

√i, (c) a = i.

4.3. Quadratic potential

The classical trajectories for this simple case were understood in detail in the context of com-plex dynamics [24] and also for curl forces [3, 15], so a brief description will suffice. Thepotential and dynamics are

U (z) =12

az2 ⇒ z = −az, (4.16)

in which a = a1 + ia2 can be any complex number. Complex curl forces correspond to a2 =0, i.e. arg a = (0, π), and, from (2.1), the curl of the force vector is constant:

C (r) = −2a2. (4.17)

The general solution is

z (t) = A+ exp(it√

a)

+ A− exp(−it

√a). (4.18)

with different orbits labelled by the complex numbers A+ and A–. Multiplying A+ and A– by acommon complex factor rotates and magnifies the trajectory, and changing |a| rescales t. Thecomplex conserved ‘energy’ (2.3) is

Ec = 2aA+A−. (4.19)

As illustrated in figure 2, the force vectors depend qualitatively on arg a. The vectors spi-ral into the origin (attracting) if − 1

2π < arg a ( mod 2π) < 12π, and spiral out (repelling)

otherwise.Positive real a (figure 2(a)) is not a curl force, and corresponds to the equal-frequency har-

monic oscillator (SHO), all of whose orbits are periodic and elliptical. For all other values of a,orbits are unbounded. For attracting a with a2 = 0—that is, for curl forces—the orbits spiralin from infinity, loop around the zero-velocity stagnation points while reversing their curva-ture, and then spiral out to infinity. This influence of the stagnation points is a well-understoodgeneral feature of complex dynamics [25]. For repelling a, the orbits curve in from infinityand out to infinity without spiralling. Imaginary a (figure 2(c)) is the pure curl case, previouslyconsidered [1] as a force with rotational symmetry. Angular momentum is not conserved forthese curl forces, because there is an angular torque; this does not violate Noether’s theorembecause the negative sign in the kinetic energy in (2.6) means that the Hamiltonian does notpossess rotational symmetry: angular momentum is not an invariant in this case. Figure 3 shows

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J. Phys. A: Math. Theor. 53 (2020) 415201 M V Berry

Figure 3. Complex orbit (4.18) under the linear curl force (4.16), for a = 1 + i, A+ =1/A– = 0.5; the black dots are the stagnation points, around which the orbit winds afterapproaching and before receding.

a typical example of a spiralling trajectory, driven by a force that spirals into the origin as infigure 2(b).

The Schrödinger equation (3.1) is(

12

(−∂2

x + ∂2y

)+

12

a1(x2 − y2)− a2xy

)ψ (r) = Eψ (r) . (4.20)

The potential possesses a saddle at r = 0, and two collinear valleys, orthogonal to which aretwo barriers; changing a1 and a2 simply rotates this structure. This Schrödinger equation isnonseparable in x and y for a2 = 0, that is, for curl forces, because of the negative sign in thekinetic energy. But from (3.3) and (3.4) it is separable in X and Y, with factor functions thatare general solutions of (3.4), expressible as parabolic cylinder functions D [21]:

ψX (X) = C1D−1/2+EX/√

a

(a1/4X

)+ C2D−1/2−EX/

√a

(ia1/4X

), (4.21)

and similarly for ψY . A convenient choice for the product solution is

Ψ (r, EX, EY) =⟨r∣∣Ψ (EX , EY)

⟩= D−1/2+EX/

√a

(a1/4 (x + iy)

)

× D−1/2+EY/√

a∗

(ia∗1/4 (x − iy)

),

(4.22)

with the sign of EY chosen so that the energy is

E = EX − EY . (4.23)

For all non-real a, that is, for all curl forces, the Stokes phenomenon [22] controlling thelarge-argument asymptotics of D [21] indicates that Ψ(r, EX , EY) grows exponentially towardsinfinity for some sectors in the x, y plane, for almost all EX, EY . The choice (4.22) ensures thatΨ does not diverge exponentially for x, y in the positive quadrant when 0 ! arg a < π/2. Thisis because the product of the leading exponentials in the asymptotics of the two D factors has

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J. Phys. A: Math. Theor. 53 (2020) 415201 M V Berry

modulus unity for all a: the product is a pure phase factor, with phase − 12

(x2 − y2

)Im

√a −

xy Re√

a.However, for EX, EY for which the arguments of D are non-negative integers, the growth

is power-law, rather than exponential, in all sectors of the x, y plane and for all a. For theseparticular solutions [21], D involves the Hermite polynomials H, and

Ψm,n (r) =⟨r∣∣Ψm,n

⟩=

1

212 (m+n)

exp(−i

(12

(x2 − y2) Im

√a + xy Re

√a))

× Hn

(a1/4√

2(x + iy)

)Hm

(a∗1/4√

2(ix + y)

),

(4.24)

This satisfies (4.19) with energy (complex for curl forces, i.e. a2 = 0)

Em,n =√

a(

m +12

)−√

a∗(

n +12

)(m = 0, 1, 2 . . . , n = 0, 1, 2 . . .) . (4.25)

For a2 = 0, the solutions (4.22), and the power-law growing modes (4.24), are quantumstates corresponding to classical curl forces. The special case of quadratic potentials that donot generate curl forces, i.e. a positive real, corresponds to the isotropic SHO, with Schrödingerequation (4.20) that is separable in x and y. The familiar orthonormal eigenstates, writtenwithout loss of essential generality for a = 1, are

ψm,n (r) =⟨r∣∣ψm,n

⟩=

exp(− 1

2

(x2 + y2

))√πm!n!2m+n

Hm (x) Hn (y) , (4.26)

with energies

Em,n = m − n. (4.27)

Note the sign, which follows from the signs in (4.20). As a consequence, these states areinfinitely degenerate in an unusual way:

Em,n = Em+k,n+k. (4.28)

But for this case a = 1 the complex-separated solutions (4.24) have exactly the same realenergies (4.27), so, as for the linear potential, the question arises of the relation between the twosets of states. As figure 4 illustrates, the states look very different. The conventional SHO statesψm,n in (4.26) (figures 4(a) and (b)) are real, and possess intersecting patterns of nodal linesassociated with the zeros of the two Hermite polynomials. By contrast, the complex-separatedstates Ψm,n (4.24) (figure 4(c)) possess isolated zeros along the x and y axes.

As with the linear potential, the relation is that any of the states |Ψm,n⟩ can be reproducedby superposing the degenerate SHO states |ψm+k,n+k⟩ with the same energy:

|Ψm,n⟩ =∞∑

k=0

cm,n,k|ψm+k,n+k⟩. (4.29)

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J. Phys. A: Math. Theor. 53 (2020) 415201 M V Berry

Figure 4. Quantum modes for the SHO, i.e. solutions of (4.20) with a2 = 0, with quan-tum numbers m = 10, n = 5, so the energy is E = 5. (a) The classical Hermite–Gausswave amplitude |ψ10,5(r)| (equation (4.26)); (b) the same state, plotted as log |ψ10,5(r)|;(c) the complex-separated mode log |Ψ10,5(r)| (equation (4.24)).

The coefficients are

cm,n,k =⟨ψm+k,n+k

∣∣Ψm,n⟩

=1√

πm!n!2m+n

∫∫dr exp

(−1

2

(x2 + y2)− ixy

)

× Hm+k (x) Hn+k (y) Hm

(1√2

(x + iy))

Hn

(1√2

(ix + y))

.

(4.30)

The double integral converges and can be evaluated explicitly for any values m, n, k. I do nothave a general form, but the first few coefficients, with the convenient scaling

cm,n,k ≡√

2π(−i)kdm,n,k, k′ = k + 1, (4.31)

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J. Phys. A: Math. Theor. 53 (2020) 415201 M V Berry

Figure 5. Approximations ΨK,0,0(r) in (4.34) for the complex mode Ψ0,0(r) as a seriesof the SHO eigenmodes ψk,k(r), around the circle r = (r cos θ, r sin θ) with r = 2,for (a) K = 2; (b) K = 5; (c) K = 20; (d) K = 100. Full red curves: Re (Ψ00 (r)) =cos

!12 r2 sin 2 θ

"; full black curves: Im (Ψ00 (r)) = − sin

!12 r2 sin 2 θ

"; dashed black

curves: approximation Re(Ψκ,00(r)); dashed red curves: approximation Im(Ψκ,00(r))

Figure 6. Families of classical trajectories with the same energy. (a) a =√

(2i), Ec =2√

(2i), A+ = 0.1(0.1)1.0, A– = 1/ A+; (b) a = 1/2, E = 1, A+ = 0.1(0.02)0.95, A– =1/ A+.

are

d0,0,k = 1, d0,1,k = 2√

k′, d1,1,k = 2(2k′ + 1

),

d0,2,k = 4√

k′2 + k′, d1,2,k = 8(k′ + 1

)√k′ + 2, d2,2,k = 8

(2k′2 + 6k′ + 5

),

d0,3,k = 8√

k′3 + 3k′2 + 2k′, d1,3,k = 8√

4k′4 + 24k′3 + 53k′2 + 51k′ + 18,

d2,3,k = 162(

k′2 + 8k′ + 9)√

k′ + 2, d3,3,k = 16(

4k′3 + 30k′2 + 8k′ + 75)

.

(4.32)

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J. Phys. A: Math. Theor. 53 (2020) 415201 M V Berry

Figure 7. Trajectories zn,m(t) (equation (4.37)) for c = −3.5(1)3.5, for (a) n = 3, (b) n= 4, (c) n = 5, (d) n = 6.

The limiting behaviour of the high coefficients in the superposition is

cm,n,k →k≫1

2m+nk12 (m+n). (4.33)

To illustrate the convergence of the series (4.29), we consider the simplest case m = n = 0,for which the sequence of approximants is

Ψ0,0 (r) = exp (−ixy)

= limK→∞

√2 exp

(−1

2

(x2 + y2)

) K∑

k=0

(−i)k

2kk!Hk (x) Hk (y) ≡ lim

K→∞ΨK,0,0 (r) .

(4.34)

Figure 5 shows how the approximants converge onto exp(−ixy) around a circler =

√(x2 + y2).

Semiclassically, the relation between the states involves families of different trajectorieswith the same energy. According to (4.19), the trajectories (4.18) with complex energy Ec for

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J. Phys. A: Math. Theor. 53 (2020) 415201 M V Berry

Figure 8. Log Ψn(x,y) (equation (4.38)) for (a) n = 0, (b) n = 1, (c) n = 2, (d) n = 3,(e) n = 4, (f) n = 5.

given a must satisfy A– = constant/A+ for different values of A+. For the complex modes (4.24)for a curl force (a2 = 0), the trajectories form a family, illustrated in figure 6(a), of the escapingtrajectories like that shown in figure 2. For a real, the trajectories are the familiar ellipses. Afamily, all with the same energy, is shown in figure 6(b); the trajectories have different eccen-tricities, with those more nearly circular being larger; this unfamiliar feature—contrasting withthe standard SHO where the larger ellipses have more energy—originates in the negative signsof the y velocity and momentum in (2.4) and (2.5).

4.4. Power-law potential, zero energy

A general power-law potential is known to have intricate classical trajectories, and few exactresults are known for the quantum counterpart. But for zero energy the trajectories and thequantum states can be described in some detail. A convenient form for this class of potentialsis

U (z) = −2zn. (4.35)

The curl of the force F (z) = −U′ (z) is, from (2.1),

C (r) = 4n (n − 1) rn−2 sin ((n − 2) θ) , (4.36)

which is non-zero for n > 2, so these potentials describe complex curl forces.In contrast to non-zero energies, for which almost all trajectories driven by curl forces

escape, the zero-energy Newtonian trajectories are bounded: between t = ±∞, they traceclosed loops. The exact solutions of (1.2) consist of branches, labelled by m:

zn,m (t, c) =Kn,m

(t − ic)2/(n−2) , Kn,m =exp

(2πim/ (n − 2)

)

(n − 2)2/(n−2) (0 ! m ! n − 3) . (4.37)

Equation (2.3) confirms that the solutions have Ec = 0. The parameter c labels the differentloops; a real parameter ic would simply correspond to changing the initial position on the loop.For different c, the loops are nested; some are shown in figure 7. The totality of loops for allreal c fills the plane.The quantum states can also be described analytically. The solutions of the complex-separatedSchrödinger equations (3.4) with EX = EY = 0 are Bessel functions [21], and the choice that

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J. Phys. A: Math. Theor. 53 (2020) 415201 M V Berry

is nonsingular at r = 0 and singlevalued under continuation is

Ψn (r) = |r| J1/(n+2)

(2

n + 2(x + iy)

12 n+1

)J1/(n+2)

(2

n + 2(x − iy)

12 n+1

)

= r∣∣∣∣J1/(n+2)

(2

n + 2r

12 n+1 exp

(i(

12

n + 1)θ

))∣∣∣∣2

·(4.38)

These states are not square-integrable: they do not represent bound states. There are lines ofzeros in the n + 2 directions θ = 2mπ/(n + 2) (0 ! m ! n + 1). Between these directions,Ψn(r) grows exponentially with r. Figure 8 shows the first few cases. For n = 0 (figure 8(a)),U(z) is a constant potential, the Bessel functions J1/2 are sines [21], and (4.38) is a zero-energyreal superposition of complex plane waves (cf. section 4.1):

Ψ0 (r) =1π

(cosh 2y − cos 2x) . (4.39)

For n = 1 (figure 8(b)), the Bessel functions J1/3 can be expressed as a sum of the Airyfunctions Ai and Bi [21], so the zero-energy solutions (4.38) fall into the class consideredin section 4.2, though different from the pure Ai states illustrated in figures 1(a) and 1(c). Forn = 2 (figure 8(c)), the Bessel functions J1/4 are superpositions of parabolic cylinder functionsD−1/2 (4.21) [21] with a = −4 (cf (4.16) and (4.35)) and EX = EY = 0, different from thefinite-energy solutions illustrated in figure 4.

Semiclassically, the large r asymptotics of the statesΨn(r) should be expressible as a densityρ(z)of the family of trajectories zn,m(t, c) in (4.37) with a weight function w(z(t, c)). The relation,involving the unexpectedly simple Jacobian, is

ρ (z) = w (z)∣∣∣∣∂ (x, y)∂ (t, c)

∣∣∣∣−1

=w (z)4|z|n . (4.40)

w(z) is fixed by the asymptotics of |Ψn(r)|2: narrow angular spikes in the growing sectors,contrasting sharply with the densities on individual trajectories.

5. Concluding remarks

The real Hamiltonian classical complex curl force interpretation explored here, and its Her-mitian quantisation, can be applied to all potentials previously studied in complex dynamics.These include a range of potentials more sophisticated than linear and quadratic, where the clas-sical trajectories have been understood in detail, including: cubic, quartic [26] and higher poly-nomial potentials; periodic potentials [8, 10, 27]; potentials with poles as well as zeros [11];and multivalued potentials [6] (where trajectories can wander between the different Riemannsheets). In some cases [10], the classical dynamics can be described using elliptic functions,whose arguments as a function of time can pass close to poles, leading to periodic trajectoriesdetermined by arithmetic conditions, and wild excursions—erratic behaviour which althoughnot chaotic in the usual sense (exponential separation of trajectories in a bounded region) is nev-ertheless sensitive to initial conditions. (Hamiltonians where the momentum p appears higherthan quadratically [11] also generate very interesting complex-plane dynamics, but do not cor-respond to curl forces (1.1), because the resulting acceleration depends on velocity as well asposition.)

The real classical Hamiltonians representing complex curl forces can all be quantisedin the conventional way, using either of the Hermitian Hamiltonians in the Schrödinger

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J. Phys. A: Math. Theor. 53 (2020) 415201 M V Berry

equations (3.1) or (3.2). Although formally conventional, the quantisation described herepossesses unfamiliar features, raising a number of questions.

• Does nonconservativeness, implied by the nonzero curl (2.1), lead to characteristicquantum behaviour?

• What are the quantum implications of the anisotropic kinetic energy in (3.1)?• For general quantised curl forces, where almost all classical trajectories are unbounded, is

it possible to create solutions of (3.1) that for real energy grow slower than exponentially,e.g. polynomially, everywhere in the x, y plane? These would represent quantum statesin the unbounded curl force potentials. They would be superpositions of the complex-separated states (3.3) with different energies EX, EY , possibly complex but whose sum isreal—complex-curl analogues of the non-curl superpositions (4.13) and (4.29); in the X,Y basis, they would be entangled.

• Can bound states exist, possibly associated with Bohr–Sommerfeld quantisation [26] ofthe isolated periodic orbits in cubic and higher potentials, even though almost all classi-cal trajectories are unbounded? (In other contexts, for example scattering by rotationallysymmetric potentials, isolated unstable closed orbits can exist; these are associated notwith bound states, but with the interesting scattering phenomenon of ‘orbiting’ [28–30].)

• Can some of the bound states studied in PT non-Hermitian quantum mechanics [16] bealternatively understood as real-energy bound states of real Hamiltonians (3.1) in the fullx, y plane, rather than in particular sectors?

• Does this study suggest an approach to the open question of quantising general classicalcurl forces, for which a Hamiltonian description seems unavailable [2]?

Although the emphasis of this paper has been mathematical—the exploration of someunusual classical and quantal formalisms—it is natural to consider applications. One pos-sibility is that the anisotropic momentum dependence of the Hamiltonians considered here((2.6) and (3.1)) could model the effects of external fields, represented by potentials satisfyingLaplace’s equation, on electrons in solids where an effective mass is negative, with confiningwalls modelled by boundary conditions (as in (4.7)).

Acknowledgments

I thank Professor E Abramochkin for help evaluating Airy integrals, and Professor P Shuklafor a helpful comment.

Appendix A. Overlap integral (4.14) and superposition (4.13)

It is convenient to use the momentum representation:

⟨ψ(Ex , Ey

) ∣∣Ψ (EX, EY)⟩

=1

2π

∫∫dk

⟨ψ(Ex , Ey

) ∣∣k⟩ ⟨

k∣∣Ψ (EX, EY)

⟩. (A.1)

For |ψ⟩ (cf (4.8)), this is derived from the integral representations of the two Ai functions, afterscaling the integration variables:

⟨ψ(Ex, Ey

) ∣∣ k⟩

=1

2π√

a1a2exp

(i(− k3

x

6a1−

k3y

6a2+

kxEx

a1+

kyEy

a2

)). (A.2)

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J. Phys. A: Math. Theor. 53 (2020) 415201 M V Berry

The counterpart for |Ψ⟩ is, from (4.12),

⟨k∣∣Ψ (EX, EY)

⟩=

12π

∫∫dr exp (−ik · r)

⟨r∣∣Ψ (EX, EY)

⟩

=1

4π

(2|a|

)2/3

× exp

(i

((kx − iky

)3

12a+

(kx + iky

)3

12a∗

)− EX (x − iy)

a

)− EY (x + iy)

a∗ .

(A.3)

This also uses the integral representations of Ai, and transformations of the double integral,using the convenient variables KX = (kx − iky)/2, KY = (kx + iky)/2 (cf (2.7)), satisfying k · r =KXX + KY Y.

The overlap (A.1) now has the form

⟨ψ(Ex , Ey

) ∣∣Ψ (EX , EY)⟩

=1

8π2√a1a2

(2|a|

)2/3 ∫∫dk exp (iΦ (k)) , (A.4)

in which Φ(k) is the sum of the two exponents in (A.2) and (A.3). After some algebra, andtransformation to the new variables

q = kxa2 + kya1, r = kxa1 − kya2, (A.5)

with Jacobian∣∣∣ ∂(q,r)∂(kx ,ky)

∣∣∣ = |a|2, the phase transforms to the degenerate cubic form

Φ (k) = − q3

6|a|2a1a2+

q

|a|2a1a2

(a2

2Ex + a21Ey + ia1a2 (EX − EY)

)

+r

|a|2(EX + EY −

(Ex − Ey

)).

(A.6)

The formula (4.14) now follows directly, with the Airy function coming from the q integrationand the delta function from the r integration.

As a check on the formal manipulations leading to the overlap (4.14), we now con-firm that the superposition (4.13) reproduces the complex-separated state (4.12) in positionrepresentation. After transforming the integration variables in (4.13) to

Ex =12

E + v, Ey = −12

E + v (A.7)

and the destination variables to

EX =12

E + u, EY =12

E − u, (A.8)

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J. Phys. A: Math. Theor. 53 (2020) 415201 M V Berry

and evaluating the E integration using the delta-function in (4.14), we need to show that

⟨r∣∣Ψ (EX , EY)

⟩=

(4

a1a2

)1/3

×∫ ∞

−∞dv Ai

((2a1)1/3

(x − E + 2v

2a1

))Ai

((2a2)1/3

(y − (−E + 2v)

a2

))

× Ai

⎛

⎜⎝ − 21/3

(|a|2a1a2

)2/3

(12

(a2

2 − a21

)E + |a|2v + 2ia1a2u

)⎞

⎟⎠ .

(A.9)

The integral has the form

L =

∫ ∞

−∞dξ Ai (aξ + b) Ai (cξ + d) Ai (eξ + f ) (A.10)

Its evaluation follows the treatment in section (3.6.3) of [26], but with a correction and slightmodification suggested by Abramochkin (private communication); therefore it is worth statingthe result explicitly.

L =

∣∣∣∣e

f 1/6

∣∣∣∣Ai((

b − a f /e)δ −

(c − a f /e

)γ

K1/6

)Ai

((c − a f /e

)α−

(b − a f /e

)β

K1/6

),

(A.11)

involving the following quantities:

K =1e6

(a6 + c6 + e6 − 2a3c3 − 2a3e3 − 2c3e3) ,

A = 1 − a3

e3 , B = −a2ce3 , C = −ac2

e3 , D = 1 − c3

e3 ,

α3 =12

(A +

1√K

(A2D − 3ABC + 2B3)

),

β3 =12

(C − 1√

K

(AD2 − 3BCD + 2C3)

),

γ3 =12

(A − 1√

K

(A2D − 3ABC + 2B3)

),

δ3 =12

(C +

1√K

(AD2 − 3BCD + 2C3)

).

(A.12)

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J. Phys. A: Math. Theor. 53 (2020) 415201 M V Berry

Explicitly, these quantities are

a = −21/3

a2/31

, b =

(x − E

2a1

)(2a1)1/3, c = −21/3

a2/32

, d =

(y +

E2a2

)(2a2)1/3,

e = −21/3|a|2/3

(a1a2)2/3 , f = −21/3

(12

(a2

2 − a21

)E + 2ia1a2u

)

(a1a2)2/3∣∣a4/3

∣∣ ,

A =a2

1

|a|2, B = −a2/3

1 a4/32

|a|2C = −a4/3

1 a2/32

|a|2, D =

a22

|a|2, K = −

(2a1a2

|a|2

)2

α =( a1

2a∗

)1/3, β = i

( a2

2a∗

)1/3γ =

( a1

2a

)1/3δ = −i

( a2

2a

)1/3.

(A.13)

Substitution into (A.11) leads to the expression (4.12) for⟨r∣∣Ψ (EX, EY)

⟩, confirming the

superposition.

ORCID iDs

M V Berry https://orcid.org/0000-0001-7921-2468

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